A remarkable property of biological systems is the formation of spatial structures. These can arise by self-organization, as a consequence of the interactions between individuals. To study these structures and how they emerge, simple models for the dynamics of the spatial density of a population, which take into account only certain elementary processes (such as reproduction, competition and dispersion) have been very useful. In particular, the FKPP (Fisher-Kolmogorov-Petrovski-Piskunov) equation, which simply includes logistic growth plus normal diffusion, is a classic model for the dynamics of a population of a single species. Within the minimalist framework of the FKPP equation and its variants, distance (or, non-local) competition is primarily responsible for producing spatial oscillations in population density. However, non-locality can also be present in other processes. Then, a first objective of this thesis is to investigate how the different spatial scales which are present in each process can interfere between them, affecting the formation of patterns in a homogeneous environment with periodic boundary conditions. For this purpose, we consider a generalization of the FKPP equation in which all terms are nonlocal. While competition is the main process behind pattern formation, we show that the other two can act constructively or destructively. For example, diffusion, which commonly homogenizes, can favor the formation of patterns depending on the format and range of the influence functions of each process. In a second study, motivated by experimental results, we seek to understand how the variability of the diffusivity can impact the spatial organization of the population inside and outside a refuge (a high-quality region immersed in a hostile environment). Therefore, we consider another generalization of the FKPP equation, with non-locality only in the intra-species competition process, modified to take into account the presence of the refuge. In addition to the spatial dependence of the growth rate, which is the main distinguishing feature of a refuge in a hostile environment, we also consider the fact that mobility can be spatially heterogeneous or depend on population density. We focus on two cases in which diffusivity responds to the density of individuals, decreasing or increasing with population density. For comparison, we also address spacedependent diffusivity, with different values inside and outside the refuge. We observed that the threshold of pattern formation in parameter space is quite robust under the presence of these heterogeneities. On the other hand, density dependence can produce a feedback that is absent in homogeneous media, and that affects the shape of the patterns. In all cases, the results were obtained by numerical simulations of the integro-differential equations and by analytical considerations.